Trigonometry Examples

$\cos (x) (\tan (x) - \sec (- x)) = \sin (x) - 1$

Step 1

Start on the left side.

$\cos (x) (\tan (x) - \sec (- x))$

Step 2

Since $\sec (- x)$ is an even function, rewrite $\sec (- x)$ as $\sec (x)$ .

$\cos (x) (\tan (x) - \sec (x))$

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

$\cos (x) (\frac{\sin (x)}{\cos (x)} - \sec (x))$

Step 3.2

Apply the reciprocal identity to $\sec (x)$ .

$\cos (x) (\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)})$

Step 4.2

Cancel the common factor of $\cos (x)$ .

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Step 4.2.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)})$

Step 4.2.2

Rewrite the expression.

$\sin (x) + \cos (x) (- \frac{1}{\cos (x)})$

Step 4.3

Cancel the common factor of $\cos (x)$ .

$\sin (x) - 1$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cos (x) (\tan (x) - \sec (- x)) = \sin (x) - 1$ is an identity